We consider navigation or search schemes on networks which have a degree distribution of the form 𝑃(𝑘) ∝ exp(−𝑘^𝛾). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter 𝜆. The searches are realistic in the sense that not all search chains can be completed. An estimate of 𝜇 = ρ/𝑠_d, where 𝜌 is the success rate and 𝑠_d the dynamic path length, shows that for a network of 𝑁 nodes, 𝜇 ∝ 𝑁^-𝛿 in general. Dynamic small world effect, i.e., 𝛿 ≃ 0 is shown to exist in a restricted region of the 𝜆 - 𝛾 plane.